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and others. Of cosingular complexes of higher degree nothing is We have practically to study the intersection of two quadrics known.

F and F' in six variables, and to classify the different cases arising Following J. Plücker, we give an account of the lines of a quadratic we make use of the results of Karl Weserstrass on the equivalence complex that meet a given line.

conditions of two pairs of quadratics. As far as at present required, The cones whose vertices are on the given line all pass through they are as follows: Suppose that the factorized form of the deter: eight fixed points and envelop a surface of the fourth degree; the minantal equation Disct (F+F')=o is conics whose planes contain the given line all lie on a surface of the fourth class and touch eight fixed planes. It is easy to see by ele.

(1-a)':+3+'3...(^-891+'z+'s+...... mentary geometry that these two surfaces are identical. Further, where the root a occurs sitsatsa... times in the determinant, the given line contains four singular points As, A:, As, A., and the $:+s:... times in every first minor, siti ... times in every second planes into which their cones degenerate are the eight common minor, and so on; the meaning of each exponent is then perfectly tangent planes mentioned above; similarly, there are lour singular definite. Every factor of the type (1-a)• is called an elementartheit planes, ai! 02, 03, go through the line, and the eight points into (elementary divisor) of the determinant, and the condition of equiva. which their conics degenerate are the eight common points above. lence of two pairs of quadratics is simply that their determinants have The locus of the pole of the line with respect to all the conics in the same elementary divisors. We write the pair of forms symbolicplanes through it is a straight line called the polar line of the given ally thus [(sı$...), (4:19 ...), ...), letters in the inner brackets one; and through this line passes the polar plane of the given line referring to the same factor. Returning now to the two quadratics with respect to each of the cones. The name polar is applied in the representing the complex,' the sum of the exponents will be six, ordinary analytical sense; any line has an infinite number of polar and two complexes are put in the same class if they have the same complexes with respect to the given complex, for the equation of the symbolical expression; i.e. the actual values of the roots of the latter can be written in an infinite number of ways; one of these determinantal equation need not be the same for both, but their polars is a straight line, and is the polar line already introduced. manner of occurrence, as far as here indicated, must be identical in The surface on which lie all the conics through a line 1 is called the the two. The enumeration of all possible cases is thus reduced Plucker surface of that line: from the known properties of (2, 2) to a simple question in combinatorial analysis, and the actual study correspondences it can be shown that the Plücker surface of I cuts

1 of any particular case is much facilitated by a useful rule of Klein's in a range of the same cross ratio as that of the range in which the for writing down in a simple form two quadratics belonging to a Plucker surface of h cuts I. Applying this to the case in which ly given class-one of which, of course, represents the equation conis the polar of I, we find that the cross ratios of (A., A2, A3, A.) and necting line coordinates, and the other the equation of the complex. (01. 03. @s. 2.) are equal. The identity of the locus of the A's with the The general complex is naturally (11); the complex of tangents envelope of the a's follows at once; moreover, a line meets the to a quadric is (11), (111)) and that of lines meeting a conic is singular surface in four points having the same cross ratio as that 1(222)). Full information will be found in Weiler's memoir, Math. of the four tangent planes drawn through the line to touch the sur

Ann, vol. vii. face. The Plücker surface has eight nodes, eight singular tangent

The detailed study of each varicty of complex opens up a vast planes, and is a double line. The relation between a line and its subject; we only mention two special cases, the harmonic complex polar line is not a reciprocal one with respect to the complex; but and the tetrahedral complex. W. Stahl has pointed out that the relation is reciprocal as far as the

The harmonic complex, first studied by Battaglini, is generated singular surface is concerned.

in an infinite number of ways by the lines cutting two quadrics to facilitate the discussion of the general quadratic complex we harmonically: Taking the most general case, and referring the

introduce Klein's canonical form. We have, in fact, to quadrics to their common self-conjugate tetrahedron, we can find its Quadratic complexes.

deal with two quadratic equations in six variables; and by equation in a simple form, and verify that this complex really

suitable linear transformations these can be reduced to the depends only on seventeen constants, so that it is not the most form

general quadratic complex. It belongs to the general type in so far Oix,? +Q2x;? +ajx;? +arxi? tants? tacxis=0

as it is discussed above, but the roots of the determinant are in inX;' + Xg? + xy + xy + x3 + x = 0

volution. The singular surface is the tetrahedroid " discussed by subject to certain exceptions, which will be mentioned later.

Caylcy: As a particular case, from a metrical point of view, we have Taking the first equation to be that of the complex, we remark perpendicular tangent planes of a quadric, the singular surface now

L. F. Painvin's complex generated by the lines of intersection of that both equations are unaltered by changing the sign of any co-being Fresnel's wave surface. The tetrahedral or Reye complex is ordinate: the geometrical meaning of this is, that the quadratic the simplest and best known of proper quadratic complexes. It is mental complexes, for changing the sign of a coordinate is equivalent generated by the linos which cut the faces of a tetrahedron in a to taking the polar of a line with respect to the corresponding cross ratio at the four vertices. The singular surface is made up of fundamental complex. It is easy to establish the existence of the faces or the vertices of the fundamental tetrahedron, and each six systems of bitangent linear complexes, for the complex edge of this tetrahedron is a double line of the complex. The hits thxo tistotletetletstluxoro is a bitangent when 12

complex was first discussed by K. T. Reye as the assemblage of lines

1,2 L?
h=0, and +


joining corresponding points in a homographic transformation of ay-da, - Q

space, and this point of view leads to many important and elegant and its lines of contact are conjugate lines with respect to the first properties. A (metrically) particular case of great interest is the lunda mental complex. We therefore infer the existence of six systems complex generated by the normals to a family of confocal quadrics, of bitangent lines of the complex, of which the first is given by and for many investigations it is convenient to deal with this com. -+ +

plex referred to the principal axes. For example, Lie has developed 'an-d'an-G'04-'a-do-Q,

the theory of curves in a Reye complex (i.e. curves whose tangents Each of these lines is a bitangent of the singular surface, which is belong to the complex) as solutions of a differential equation of the therefore completely determined as being the local surface of the form (6-c)ądydz+(-a)ydzdx + (a-b)zdxdy = 0, and we can simplify (2, 2) congruence above. It is thence easy to ver that the two this equation by a logarithmic transformation. Many theorems complexes Ears -o and Ebxo = are cosingular if b. = 2,5+wa,vtp. connecting.complexes with differential equations have been given

The singular surface of the general quadratic complex is the by Lie and his school. A line complex, in fact, corresponds to a famous quartic, with sixteen nodes and sixteen singular tangent Mongian equation having coi line integrals. planes, first discovered by E. E. Kümmer.

As the coordinates of a line belonging to a congruence are functions We cannot give a full account of its properties here, but we deduce of two independent parameters, the theory of congruences is analogous at once from the above that its bitangents break up into six (2, 2) to that of surfaces, and we may regard it as a fundamental congruences, and the six linear complexes containing these are inquiry to find the simplest form of surface into which

Coogru. mutually in involution. The nodes of the singular surface are points a given congruence can be transformed. Most of those

ences. • whose complex cones are coincident planes, and the complex conic whose properties have been extensively discussed can be represented in a singular tangent plane consists of two coincident points. This on a plane by a birational transformation. But in addition to the configuration of sixteen points and planes has many interesting difficulties of the theory of algebraic surfaces, a subject still in its properties; thus each plane contains six points which lie on a conic, infancy, the thcory of congruences has other difficulties in that a while through each point there pass six planes which touch a quadric congruence is seldom completely represented, even by two equations.

In many respects the Kümmer quartic plays a part in three A fundamental theorem is that the lines of a congruence are in dimensions analogous to the general quartic curve in two; it further general bitangents of a surface; in fact, since the condition of intergives a natural representation of certain relations between hyper- section of two consecutive straight lines is ld. + dmdu + dndvæo, a elliptic functions (cf. R. W. H. T. Hudson, Kummer's Quartic, 1905). line 1 of the congruence meets two adjacent lines, say, b, and h.

As might be expected from the magnitude of a form in six variables, Suppose I, 1, lie in the plane pencil (Aja,) and I, 12 in the plane pencil the number of projectivally distinct varieties of quadratic complexes (A:0), then the locus of the A's is the same as the envelope of the

is very great; and in fact Adolf Weiler, by whom the a's, but az is the tangent plane at A, and Q, at Az. This surface is Classic

question was first systematically studied on lines indicated called the focal surface of the congruence, and to it all the lines I

by Klein, enumerated no fewer than lorty-nine different are bitangent. The distinctive property of the points A is that two quadratke

týpes. But the principle of the classification is so im- of the congruence lines through them coincide, and in like manner complexes.

portant, and withal so simple, that we give a briel sketeh the planes a each contain two coincident lines. The focal surface which indicates its essential features.

consists of two sheets, but one or both may degenerate into curves;


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thus, for example, the normals to a surface are bitangents of the

VI. Non-EUCLIDEAN GEOMETRY surface of centres, and in the case of Dupin's cyclide this surface degenerates into two conics.

The various metrical geometries are concerned with the in the discussion of congruences it soon becomes necessary to properties of the various types of congruence-groups, which are introduce another number 7, called the rank, which expresses the defined in the study of the axioms of geometry and of their number of plane pencils each of which contains an arbitrary line and two lines of the congruence. The order of the focal surface is immediate consequences. But this point of view of the subject 2m(n-1)-21, and its class is m(m-1)-2r. Our knowledge of is the outcome of recent research, and historically the subject congruences is almost exclusively confined to those in which either has a different origin. Non-Euclidean geometry arose from the m or n does not exceed two. We give a brief account of those of discussion, extending from the Greek period to the present day, being especially interesting. A congruence generally has singular of the various assumptions which are implicit in the traditional points through which an infinite number of lines pass; a singular Euclidean system of geometry. In the course of these investigapoint is said to be of order r when the lines through it lie on a cone tions it became evident that metrical geometries, each internally of the rth degree. By means of formulae connecting the number of consistent but inconsistent in many respects with each other gruence Kummer proved that the class cannot exceed seven. The and with the Euclidean system, could be developed. A short local surface is of degree four and class 2m; this kind of quartic historical sketch will explain this origin of the subject, and surface has been extensively studied by Kümmer, Cayley, Rohn and describe the famous and interesting progress of thought on the others. The varieties (2, 2), (2, 3), (2, 4), (2, 5) all belong to at subject. But previously a description of the chief characteristic of (2,6) congruences which includes all the above as special cases. properties of elliptic and of hyperbolic geometries will be given, The congruence (2, 2) belongs to a linear complex and forty different assuming the standpoint arrived at below under VII. Axioms Reye complexes; as above remarked, the singular surface is of Geometry. Kümmer's sixteen-nodal quartic, and the same surface is focal for

First assume the equation to the absolute (cf. loc. cit.) to six different congruences of this variety. The theory of (2, 2) congruences is completely analogous to that of the surfaces called be w- x? – ye --=0. The absolute is then real, and the cyclides in three dimensions. Further particulars regarding quad. geometry is hyberbolic. ratic congruences will be found in Kümmer's memoir of 1866, and The distance (dz) between the two points (I1, Yr, 21, 2) and (x, y, the second volume of Sturm's treatise. The properties of quadratic 2, ws) is given by congruences having singular lines, i.e. degenerate focal surfaces, are

cosh (dizly) = (w/w. – 3183-919-22)/{(w' - 28-4-2) not so interesting as those of the above class; they have been

(wy - x - yz - 27")} discussed by Kümmer, Sturm and others. Since a ruled surface contains only as elements, this theory is

The only points to which the metrical geometry applies are those practically the same as that of curves. If a linear complex contains

within the region enclosed by the quadric; the other points are Ruled more than n generators of a ruled surface of the nth degree, 11*+my+ni2tr,w=o and 12x +my+n:+1w=, is given by

improper ideal points." The angle (0:1) between two planes, surfaces.

it contains all the generators, hence for 1 = 2 there are
three linearly independent complexes, containing all the

cos 0.2 = ( +1193 - 7192)/111,?+ni+nani? generators, and this is a well-known property of quadric surfaces.

(142 +mga+nge - 17") }! (2) In ruled cubics the generators all meet two lines which may, or inay

These planes only have a real angle of inclination if they possess a not coincide; these two cases correspond to the two main classes of

line of intersection within the actual space, i.e. if they intersect. cubics discussed by Cayley and Cremona. As regards ruled quartics, Planes which do not intersect possess a shortest distance along a line the generators must lie in one and may lie in two linear complexes which is perpendicular to both of them. If this shortest distance is The first class is equivalent to a quartic in four dimensions and is

817. we have always rational, but the latter class has to be subdivided into the cosh (8.2/7) = (liha+mima +1192-917:)/{(42+m: +1° -72) elliptic and the rational, just like twisted quartic curves. A quintic

(1;2+m78 +ng? -77) skew may not lic in a linear complex, and then it is unicursal, while of Thus in the case of the two planes one and only one of the two, a sextics we have two classes not in a linear complex, viz. the elliptic and 812, is real. The same considerations hold for coplanar straight variety, having thirty-six places where a linear complex contains lines (see VII. Axioms of Geometry). Let o (fig. 67) be the point six consecutive generators, and the rational, having six such (0, 0, 0, 1), OX the line y=0, places.

:=0, OY the line 2=0, x=0, and The general theory of skews in two linear complexes is identical Oz the line x = 0, y = 0. These are with that of curves on a quadric in three dimensions and is known. the coordinate axes and are at But for skews lying in only one linear complex there are difficulties; right angles to each other.

Let the curve now lies in four dimensions, and we represent it in three by P be any point, and let o be the stereographic projection as a curve meeting a given plane in n point's distance OP, o the angle Poz, and on a conic. To find the maximum deficiency for a given degree would the angle between the planes probably be difficult, but as far as degree eight the space-curve ZOX and ZOP.

Then the cotheory of Halphen and Nother can be translated into line geometry ordinates of P can be taken to be at once. When the skew does not lie in a linear complex at all the sinh (p/4) sin cos , sinh (ply) sin e theory is more difficult still, and the general theory clearly cannot advance until further progress is made in the study of twisted

sine, sinh 9/1) cose, cosh (9/1).

II ABC is a triangle, and the REFERENCES.-The earliest works of a general nature are Plücker, sides and angles are named accordNeue Geometrie des Raumes (Leipzig, 1868); and Kummer, “ Übering to the usual convention, we have

FIG. 67. die algebraischen Strahlensysteme," Berlin Academy (1866). System

sinh (e|v)/sin A=sinh (b/x)/sin B=sinh (c(x)/sin C, (4) atic development on purely, synthetic lines will be found in the three volumes of Sturm, Liniengeometrie (Leipzig, 1892, 1893, 1896); and also vol. i. deals with the linear and Reye complexes, vols. ii. and ii. cosh (a/7) = cosh (0/7) cosh (c/v)- sinh (6/7) sinh (c/y) cos A. (5) with quadratic congruences and complexes respectively. For a highly suggestive review by Gino Loria see Bulletin des sciences

with two similar equations. The sum of the three angles of a triangle mathématiques (1893, 1897). A shorter treatise, giving a very

is always less than two right angles. The area of the triangle ABC interesting account of Klein's coordinates, is the work of Koenigs, and the vertex A moves in the fixed plane ABC so that the area

is a'r-A-B-C). II the base BC of a triangle is kept fixed La Géoméirie réglée et ses applications (Paris, 1898). English treatises are C. M. Jessop, Treatise on the Line Complex (1903); R. W. H. T.

ABC is constant, then the locus of A is a line of equal distance from Hudson, K'ümmer's Quartic (1905). Many references to memoirs on

BC. This locus is not a straight line. The whole theory of similarity

is inapplicable; two triangles are either congruent, or their angles line geometry, will be found in Hagen, Synopsis der, höheren Mathemalik, ii. (Berlin, 1894); Loria, Il passato ed il presenle delle principali determined when its three angles are

are not equal two by two. Thus the elements of a triangle are teorie geometriche (Milan, 1897); a clear résumé of the principal results is contained in the very clegant volume of Pascal, Repertorio line BC fixed, but by making C move

given. By keeping A and B and the di mathematiche superiori, ji. (lilan, 1900). Another treatise dealing off to infinity along BC, the lines BC formationen (Leipzig, 1896). Many memoirs on the subject have and AC become parallel, and the sides

a and b become infinite. Hence from appeared in the Mathematische Annalen; a full list of these will be found in the index to the first fifty volumes, p. 115. Perhaps the equation (5) above, it follows that two two memoirs which have left most impression on the subsequent Geometry) must be considered as making a zero angle with each

FIG. 68. complexe des ersten und zweiten Grades," Math. Ann. ii.; and Lie, other.. Also if

. B be a right angle, from the equation (5), remem“ Uber Complexe, insbesondere Linien- und Kugelcomplexe,"

bering that, in the limit, Math. Ann. V.

U. H. GR.)

cosh (0/7)/cosh (b/y) = cosh (a/)/sinh (6/7) = 1,





we have
cos A=tanh (c(2)

(6). by the plane p, but P and R are not separated by P, nor are Q The angle A is called by N. I. Lobatchewsky the " angle of parallel- and R.

Let A, B, C be any three non-collinear points, then four triangles The whole theory of lines and planes at right angles to each other are defined by these points. Thus is a, b, c and A, B, C are the is simply the theory of conjugate elements with respect to the elements of any one triangle, then the four triangles have as their absolute, where ideal lines and planes are introduced.

elements: Thus if I and I' be any two conjugate lines with respect to the


A, B. C. absolute (of which one of the two must be improper, say I'), then

ty-b, ry-6,

A, -B, 1-C. any plane through l' and containing proper points is perpendicular

(3) Tya, b, ry-, -A, B 1-C. to I. Also if p is any plane containing proper points, and P is its

ty-a, ay-b,

7-A, 1-B, C. pole, which is necessarily improper, then the lines through P are the normals to P. The equation of the sphere, centre (x2, 91, 31, w) The formulae connecting the elements are and radius p, is

sin A/sin (4/2) = = sin B/sin (b/n) = sin C/ain (cy), . (12) (w,*-**-3-34) (wi-ri — 22-2) cosh”(pły) =

and (ww-xix-y13-3,2), (7).

cos (a/y)=cos (617) cos (c/y)+sin (6/7) sin (chy) cos A, (13) The equation of the surface of equal distance (o) from the plane with two similar equations. Ix+my+na+rw=o is (+m+gr) (uê --y-g) sinh?(g/x)=

Two cases arise, namely (I.) according as one of the four triangles

has as its sides the shortest segments between the angular points,

(rw+lxt-my+nz) (8). or (II.) according as this is not the case. When case I. holds there A surface of equal distance is a sphere whose centre is improper: within a sphere of radius oy only case I. can hold, and the principal

is said to be a “principal triangle."? If all the figures considered lie and both types of surface are included in the family

triangle is the triangle wholly within this sphere, also the peculiarities k?(W2 - X - 72 – 8*) = (ar+by+ca+dw)? . . (9). in respect to the separation of points by a plane cannot then arise. But this family also includes a third type of surfaces, which can

The sum of the three angles of a triangle ABC is always greater than be looked on either as the limits of spheres whose centres have Thus as in hyperbolic geometry the theory of similarity does not

two right angles, and the area of the triangle is g?(A+B+C--). approached the absolute, or as the limits of surfaces of equal distance hold, and the elements of a triangle are determined when its three whose central planes have approached a position tangential to the angles are given. The coordinates of a point can be written in the absolute. These surfaces are called limit-surfaces. Thus (9) denotes

form a limit-surface, if & -a?-2-(?=0. Two limit-surfaces only differ in position. "Thus the two limit-surfaces which touch the plane where p. 0 and have the same meanings as in the corresponding

sin (ply) sin 0 cos o, sin (ply) sin 0 sin , sin (p/V) cos 0, cos (ply), YOZ at o, but have their concavities turned in opposite directions, formulae in hyperbolic geometry. Again, suppose a watch is laid have as their equations

on the plane OXY, face upwards with its centre at 0, and the line 22- y2 – 3: =(w+x).

12 to 6 (as marked on dial) along the line YOY. Let the watch be The geodesic geometry of a sphere is elliptic, that of a surface of continually pushed along the plane along the line ox, that is, in equal distance is hyperbolic, and that of a limit-surface is parabolic the direction 9 to 3. Then the line XOX being of finite length, the (1.e. Euclidean). The equation of the surface (cylinder) of equal / watch will return to O, but at its first return it will be found to be distance (s) from the line OX is

face downwards on the other side of the plane, with the line 12 to 6

reversed in direction along the line YOY. This peculiarity was first () tanh?(8/1)-y-3=0.

pointed out by Felix Klein. The theory of parallels as it exists in This is not a ruled surface. Hence in this geometry it is not possible hyperbolic space has no application in elliptic geometry. But for two straight lines to be at a constant distance from each other.

another property of Euclidean parallel lines holds in elliptic geoSecondly, let the equation of the absolute be r?+y? ++++ tion of the surface (cylinder) of equal distance (8) from the line

metry, and by the use of it parallel lines are defined. For the equaW = 0. The absolute is now imaginary and the geometry is XOX is elliptic.

(r? +w*) tan (8/1)-(y2 +22) = 0. The distance (dia) between the two points (22. Yu, 21, wy) and

This is also the surface of equal distance, try-8, from the line (x2. Y?. 3, ) is given by

conjugate to XOX. Now from the form of the above equation this cos (daly) = + (x1x2+yın: +2183+w6)/{(x;?+y+z;" +10)

is a ruled surface, and through every point of it two generators pass. (x3? +372 +2° +w;?)?! (10).

But these generators are lines of equal distance from XOX. Thus Thus there are two distances between the points, and if one is diz, hroughout every point of space two lines can be drawn which are the other is ay-d12. Every straight line returns into itsell, forming lines of equal distance from a given line l. This property was disa closed series. Thus there are two segments between any two

covered by W. K. Clifford. The two lines are called Clifford's right points, together forming the whole line which contains them; one

and left parallels to l through the point. This property of paralleldistance is associated with one segment, and the other distance with ism is reciprocal, so that if m is a lelt parallel tol, then I is a left the other segment. The complete length of every straight line is ry. parallel to m. Note also that two parallel lines I and m are not

The angle between the two planes lix+my+nietrw=o and coplanar.. Many of those properties of Euclidean parallels, which do 19x+may+nzstraw=o is

not hold for Lobatchewsky's parallels in hyperbolic geometry, do

hold for Clifford's parallels in elliptic geometry. The geodesic cos 0:2 = (1,17+mimitning trips)/{(1,2+m?+n?,,?)

geometry of spheres is elliptic, the geodesic geometry of surfaces of

(I +m+x+t)}} ( ). equal distance from lines (cylinders) is Euclidean, and surfaces of The polar plane with respect to the absolute of the point (x1, Yr, 21. w) revolution can be found of which the geodesic geometry is hyperis the real plane x,x+yıy+z2+w.w=0, and the pole of the plane bolic. But it is to be noticed that the connectivity of these surfaces lix+my+nzthw=o is the point (1. ma, ni

,,). Thus from is different to that of a Euclidean plane. For instance there are only equations 10 and 11) it follows that the angle between the polar 02 congruence transformations of the cylindrical surfaces of equal plancs of the points (x1,...) and (x2,..) is duzly, and that the distance into themselves, instead of the color the ordinary plane.

which these distance between the poles of the planes (...) and (13,...) is It would obviously be possible to state "axioms 2012. Thus there is complete reciprocity between points and planes geodesics satisfy, and thus to define independently, and not as loci, in respect to all properties. This complete reign of the principle quasi-spaces of these peculiar types. The existence of such Euclidean of duality is one of the great beauties of this geometry. The theory quasi-geometries was first pointed out by Clifford.* of lines and planes at right angles is simply the theory of conjugate In both elliptic and hyperbolic geometry the spherical elements with respect to the absolute. A tetrahedron sell-conjugate geometry, i.e. the relations between the angles formed by lincs with respect to the absolute has all its intersecting elements (edges and planes passing through the same point, is the same as the and planes) at right angles. land l' are two conjugate lines, the planes through one are the planes perpendicular to the other. II

" spherical trigonometry" in Euclidean geometry. The constant P is the pole of the plane p. the lines through P are the normals to 7, which appears in the formulae both of hyperbolic and elliptic the plane p. The distance from P to p is fry. Thus every sphere geometry, does not by its variation produce dificrent types of is also a surface of equal distance from the polar of its centre, and conversely. A plane does not divide space; for the line joining any

geometry. There is only one type of elliptic geometry and one two points P and Q only cuts the plane once, in L say, then it is type of hyperbolic geometry; and the magnitude of the constant always possible to go from P to Q by the segment of the line PQ r in each case simply depends upon the magnitude of the arbitrary which does not contain L. But PandQ may be said to be separated unit of length in comparison with the natural unit of length by a plane p, is the point in which Pð cuts p lies on the shortest segment between P and Q. With this sense of " separation," it is

* CI. A. N. Whitehead, loc. cit. possible to find three points P, Q, R such that Pand are separated Çf. A. N. Whitehead, " The Geodlesic Geometry of Surfaces in

non-Euclidean Space," Proc. Lond. Math. Soc. vol. xxix. 1C1. A. N. Whitehead, Universal Algebra, Bk. vi. (Cambridge, .C1. Klein, “Zur nicht-Euklidischen Geometrie," Math. Annal. 1898).

vol. xxxvii.

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Euclideaa geometry

which each particular instance of either' geometry presents. | equal. The first hypothesis is that these are both right angles; The existence of a natural unit of length is a peculiarity common the second, that they are both obtuse; and the third, that they both to hyperbolic and elliptic geometries, and ditferentiates are both acute. Many of the results afterwards obtained by them from Euclidean geometry. It is the reason for the failure Lobatchewsky and Bolyai are here developed. Saccheri fails of the theory of similarity in them. If y is very large, that is, to be the founder of non-Euclidean geometry only because he if the natural unit is very large compared to the arbitrary unit, does not perceive the possible truth of his non-Euclidean hypoand is the lengths involved in the figures considered are not large theses. compared to the arbitrary unit, then both the elliptic and Some advance is made by Johann Heinrich Lambert in his hyperbolic geometries approximate to the Euclidean. For from Theorie der Parallellinien (written 1766; posthumously published formulae (4) and (5) and also from (12) and (13) we find, after 1786). Though he still believed in the necessary

Lambert. retaining only the lowest powers of small quantities, as the truth of Euclidean geometry, he confessed that, in formulae for any triangle ABC,

all his attempted proofs, something remained undemonstrated. c/ sin A=b/ sin B=c/ sin C,

He deals with the same three hypotheses as Saccheri, showing and

that the second holds on a sphere, while the third would hold on Q? = b2+c-2bc cos A,

a sphere of purely imaginary radius. The second hypothesis with two similar equations. Thus the geometries of small he succeeds in condemning, since, like all who preceded Bernhard figures are in both types Euclidean.

Riemann, he is unable to conceive of the straight line as finite History.—“ In pulcherrimo Geometriae corpore,” wrote Sir and closed. But the third hypothesis, which is the same as Henry Savile in 1621, “duo sunt naevi, duae labes nec quod Lobatchewsky's, is not even professedly reluted. Theory of sciam plures, in quibus eluendis et emaculendis cum Non-Euclidean geometry proper begins with Karl Friedrich parallels veterum tum recentiorum . . . vigilavit industria." Gauss. The advance which he made was rather philosophical before These two blemishes are the theory of parallels and than mathematical: it was he (probably) who first

Three the theory of proportion. The “ industry of the recognized that the postulate of parallels is possibly periods al moderns," in both respects, has given rise to important branches false, and should be empirically tested by measuring noo. of mathematics, while at the same time showing that Euclid the angles of large triangles. The history of nonis in these respects more free from blemish than had been Euclidcan geometry has been aptly divided by Felix previously credible. It was from endeavours to improve the Klein into three very distinct periods. The first-which contains theory of parallels that non-Euclidean geometry arose; and only Gauss, Lobatchewsky and Bolyai-is characterized by its though it has now acquired a far wider scopc, its historical synthetic method and by its close relation to Euclid. The origin remains instructive and interesting. Euclid's “axiom attempt at indirect proof of the disputed postulate would seem of parallels " appears as Postulate V. to the first book of his to have been the source of these three men's discoveries; but Elements, and is stated thus, “ And that, if a straight line falling when the postulate had been denied, they found that the results, on two straight lines make the angles, internal and on the same so far from showing contradictions, were just as self-consistent side, less than two right angles, the two straight lines, being as Euclid. They inferred that the postulate, is true at all, can produced indefinitely, meet on the side on which are the only be proved by observations and measurements. Only one angles less than two right angles.” The original Greek is kind of non-Euclidean space is known to them, namely, that και εάν είς δύο ευθείας ευθεία εμπίπτουσα τας εντός και επί τα which is now called hyperbolic. The second period is analytical, αυτά μέρη γωνίας δύο ορθών ελάσσονας ποιή, εκβαλλομένας τας | and is characterized by a close relation to the theory of surfaces. δύο ευθείας επ' άπειρον συμπίπτειν, εφ' & μέρη είσιν αι των δύο It begins with Riemann's inaugural dissertation, which regards όρθων ελάσσονες. .

space as a particular case of a manifold; but the characteristic To Euclid's successors this axiom had signally failed to appcar standpoint of the period is chiclly emphasized by Eugenio self-evident, and had failed cqually to appear indemonstrable. Beltrami. The conception of measure of curvature is extended Without the use of the postulate its converse is proved in Euclid's by Riemann from surfaces to spaces, and a new kind of space, 28th proposition, and it was hoped that by further efforts the finite but unbounded (corresponding to the second hypothesis postulate itself could be also proved. The first step consisted of Saccheri and Lambert), is shown to be possible. As opposed in the discovery of equivalent axioms. Christoph Clavius in to the second period, which is purely metrical, the third period 1574 deduced the axiom from the assumption that a linc whose is essentially projective in its method. It begins with Arthur points are all equidistant from a straight line is itself straight. Cayley, who showed that metrical properties are projective John Wallis in 1663 showed that the postulate follows from the properties relative to a certain fundamental quadric, and that possibility of similar triangles on different scales. Girolamo different geometries arise according as this quadric is real, Saccheri (1733) showed that it is sufficient to have a single imaginary or degenerate. Klein, to whom the development of triangle, the sum of whose angles is two right angles. Other Cayley's work is due, showed further that there are two forms equivalent forms may be obtained, but none shows any essential of Riemann's space, called by him the elliptic and the spherical. superiority to Euclid's. Indeed plausibility, which is chiefly Finally, it has been shown by Sophus Lie, that is figures are to be aimed at, becomes a positive demerit where it conceals a real freely movable throughout all space in 6 ways, no oiber assumption.

three-dimensional spaces than the above four are possible. A new method, which, though it failed to lead to the desired Gauss published nothing on the theory of parallels, and it goal, proved in the end immensely fruitful, was invented by was not generally known until after his death that he had

Saccheri, in a work entitled Euclides ab omni nacro interested himself in that theory from a very early Saccheri.

vindicatus (Milan, 1733). If the postulate of parallels date. In 1799 he announces that Euclidean geometry is involved in Euclid's other assumptions, contradictions must would follow from the assumption that a triangle can be drawn emerge when it is denied while the others are maintained. This greater than any given triangle. Though unwilling to assume led Saccheri to attempt a reductio ad absurdum, in which he this, we find him in 1804 still hoping to prove the postulate of mistakenly believed himself to have succeeded. What is interest - parallels. In 1830 he announces his conviction that geometry ing, however, is not his fallacious conclusion, but the non- is not an a priori science; in the following year he explains that Euclidean results which he obtains in the process. Saccheri non-Euclidean geometry is free from contradictions, and that, distinguishes three hypotheses (corresponding to what are now in this system, the angles of a triangle diminish without limit known as Euclidean or parabolic, elliptic and hyperbolic geo- when all the sides are increased. He also gives for the metry), and proves that some one of the three must be univer sally true. His three hypotheses are thus obtained: equal | Engel, Theorie der Parallellinien von Euklid bis auf Gauss (Leipzig;

1 On the theory of parallels before Lobatchewsky, see Stäckel und perpendiculars AC, BD are drawn from a straight line AB,

1895). The foregoing remarks are based upon the materials collected and CD are joined. It is shown that the angles ACD, BDC are in this work.


circumference of a circle of radius r the formula ak(e)k-1), The works of Lobatchewsky and Bolyai, though known and where k is a constant depending upon the nature of the space. In valued by Gauss, remained obscure and ineffective until,in 1866, 1832, in reply to the receipt of Bolyai's Appendix, he gives an they were translated into French by J. Hoüel. But

Riemann, elegant proof that the amount by which the sum of the angles of a at this time Riemann's dissertation, Über dic Hypothesen, triangle falls short of two right angles is proportional to the area welche der Geomelrie zu Grunde liegen,' was already about to be of the triangle. From these and a few other remarks it appears published. In this work Riemann, without any knowledge of that Gauss possessed the foundations of hyperbolic geometry, his predecessors in the same field, inaugurated a far more profound which he was probably the first to regard as perhaps true. It discussion, based on a far more general standpoint; and by is not known with certainty whether he influenced Lobatchewsky | its publication in 1867 the attention of mathematicians and and Bolyai, but the evidence we possess is against such a view.' philosophers was at last secured. (The dissertation dates from

The first to publish a non-Euclidean geometry was Nicholas 1854, but owing to changes which Ricmann wished to makc in it, Lobatchewsky, professor of mathematics in the new university it remained unpublished until after his death.)

of Kazan. In the place of the disputed postulate Riemann's work contains two fundamental conceptions, that Lobatchewsky.

he puts the following: “All straight lines which, in of a manifold and that of the measure of curvature of a continuous

a plane, radiate from a given point, can, with respect manifold possessed of what he calls flatness in the smallest parts. to any other straight line in the same plane, be divided into By means of these conceptions space is made to appear two classes, the intersecting and the non-intersecting. The at the end ofagradual series of more and more specialized Definition boundary line of the one and the other class is called parallel conceptions. Conceptions of magnitude, he explains, sold. to the given line." It follows that there are two parallels to the are only possible where we have a general conception given line through any point, cach meeting the line at infinity, capable of determination in various ways. The manifold consists like a Euclidean parallel. (Hence a line has two distinct points of all these various determinations, cach of which is an element at infinity, and not one only as in ordinary geometry.) The of the manifold. The passage from one clement to another may two parallels to a line through a point make equal acute angles be discrete or continuous; the manifold is called discrete or with the perpendicular to the line through the point. If p be continuous accordingly. Where it is discrete two portions of the length of the perpendicular, either of these angles is denoted it can be compared, as to magnitude, by counting; where by Il(p). The determination of II(p) is the chief problem (cf. continuous, by measurement. But measurement demands equation (6) above); it appears finally that, with a suitable superposition, and consequently some magnitude independent choice of the unit of length,

of its place in the manifold. In passing, in a continuous manifold, tan (D)=e-.

from one element to another in a determinate way, we pass Before obtaining this result it is shown that spherical trigono- through a series of intermediate terms, which form a onemetry is unchanged, and that the normals to a circle or a sphere dimensional manifold. If this whole manifold be similarly still pass through its centre. When the radius of the circle or

caused to pass over into another, each of its elements passes sphere becomes infinite all these normals become parallel, but the through a one-dimensional manifold, and thus on the whole circle or sphere does not become a straight line or .plane. It

a two-dimensional manifold is generated. In this way we can becomes what Lobatchewsky calls a limit-line or limit-surface. proceed to n dimensions. Conversely, a manifold of n dimensions The geometry on such a surface is shown to be Euclidean, limit can be analysed into one of one dimension and one of (1-1) lines replacing Euclidean straight lines. (It is, in fact, a surface dimensions. By repetitions of this process the position of an of zero measure of curvature.) By the help of these propositions clement may be at last determined by n magnitudes. We may Lobatchewsky obtains the above value of n(P), and thence the here stop to observe that the above conception of a manifold solution of triangles. He points out that his formulae result is akin to that due to Hermann Grassmann in the first edition from those of spherical trigonometry by substituting ia, ib, ic, (1847) of his Ausdehnungslehre. for the sides a, b, c.

Both concepts have been elaborated and superseded by the John Bolyai, a Hungarian, obtained results closely correspond modern procedure in respect to the axioms of geometry, and by ing to those of Lobatchewsky. These he published in an appendix the conception of abstract geometry involved therein.

Measure of to a work by his father, entitled Appendix Scientiam Riemann proceeds to specialize the manifold by con- curvature. Boly al

spalii absolute veram exhibens: a veritate cul falsitate siderations as to measurement. If measurement is to Ariomatis XI. Euclidei (a priori haud unquam decidenda) in- be possible, some magnitude, we saw, must be independent of dependentem: adjecia ad casum falsilatis, quadralura circuli position; let us consider manifolds in which lengths of lines are geometrica." This work was published in 1831, but its conception such magnitudes, so that every line is measurable by every dates from 1823. It reveals a profounder appreciation of the other. The coordinates of a point being X1, X2, . . . Xn, let us conimportance of the new ideas, but otherwise differs little from fine ourselves to lines along which the ratios dxı:drz:. . . dxn Lobatchewsky's. Both men point out that Euclidean geometry

alter continuously. Let us also assume that the element of as a limiting case of their own more general system, that the length, ds, is unchanged (to the first order) when all its points geometry of very small spaces is always approximately Euclidean, undergo the same infinitesimal motion. Then is all the increments that no a priori grounds exist for a decision, and that observation dx be altered in the same ratio, ds is also altered in this ratio. can only give an approximate answer. Bolyai gives also, as bis Hence ds is a homogeneous function of the first degree of the title indicates, a geometrical construction, in hyperbolic space, increments dr. Moreover, ds must be unchanged when all the for the quadrature of the circle, and shows that the area of the dx change sign. The simplest possible casc is, therefore, that in greatest possible triangle, which has all its sides parallel and all which ds is the square root of a quadratic function of the dx. its angles zero, is mrz?, where į is what we should now call the This case includes space, and is alone considered in what follows. space-constant.

It is called the case of fatness in the smallest parts. Its further See Stăckel und Engel, op. cit., and "Gauss, die beiden Bolyai, discussion depends upon the measure of curvature, the second und die nicht-Euklidische Geometrie." Math. Annalen, Bd. xlix.; of Riemann's fundamental conceptions. This conception, derived also Engel's translation of Lobatchewsky (Leipzig, 1898), pp. 378 fi. from the theory of surfaces, is applied as follows. Any one of

? Lobatchewsky's works on the subject are the following: On the shortest lines which issue from a given point(say the origin) the Foundations Geometry," Kazan Messenger, 1829-1830; is completely determined by the initial ratios of the dr. Two ** New Foundations of Geometry, with a complete Theory of Parallels," Proceedings of the University of Kazan, 1835 (both in such lines, defined by dx and ox say, determine a pencil, or oneRussian, but translated into German by Engel, Leipzig, 1898); dimensional series, of shortest lines, any one of which is defined * Géométrie imaginaire," Crelle's Journal, 1837; Theorie der Parallellinien (Berlin, 1840; 2nd ed., 1887; translated by Halsted, · Abhandlungen d. Königl. Ges. d. Wiss. zu Göttingen, Bd. xiii.; Austin, Texas, 1891). His results appear to have been set forth in a Ges. math. Werbe, pp. 254-269; translated by Clifford, Collected paper (now lost) which he read at Kazañ in 1826.

Mathematical Papers. Translated by Halsted (Austin, Texas, 4th ed., 1896).

*Cf. Gesamm. math. und phys. Werke, vol. i. (Leipzig, 1894).

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